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\begin{table}[htbp] 
  \caption{\bf Pricing Performance:  Out-of-Sample -  Expected Returns  ($T=24$)}
\medskip
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccccccccc}
     \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA  & - &  0.314 & 0.351 & 0.384 &  0.289  &  0.292  &  0.288  &  0.278  & 0.274 & 0.281  &  0.329 \\
  local-PCA (bal) &  0.498  &   0.497 & 0.504 &  0.501 &  0.508 &  0.512 &  0.518 &  0.521 &  0.523 &  0.525 & 0.529   \\
local-PCA (unbal) &    0.851 &  0.850 &  0.852  &  0.852 &  0.853 &  0.854 &  0.855 &  0.856 &  0.856 &  0.857 &  0.857  \\ \hline
   \multicolumn{12}{c}{In-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
   \\ \hline
  constant-PCA &   - &  0.328 & 0.290 &  0.340 &  0.476  &  0.490  &  0.503 &  0.517 &  0.518  & 0.532  & 0.536  \\
 local-PCA (bal) & 0.642 &   0.449  & 0.548 &  0.599 &  0.642 &  0.690 &  0.734 &  0.754 &  0.770 & 0.781 &  0.792  \\
 local-PCA (unbal) &   0.879 &  0.817 &  0.847 &  0.860 &  0.871 &  0.883 &  0.893 & 0.901 &  0.906 &  0.912 &  0.918 \\  \hline 
    \end{tabular}
        \parbox{6.3in}{\footnotesize{The table shows the cross-sectional correlations between the average returns and the model-implied expected return estimated by 
        constant-PCA (first row), local-PCA balanced (second row), and local-PCA unbalanced (third row), when the number of risk factors ranges from one to $10$  (columns $3-12$), with the second column referring to the
      (time-varying) estimated number of factors $\tilde{r}$, described in Section~\ref{emp_number_tv}.
       The average returns and model-implied expected returns are first evaluated over every rolling window of size $T=24$ and then averaged across all rolling windows. 
 Data are monthly from Jan 1980 until Dec 2019.}}
\label{Table2OAT24}
\end{table}

 
  \begin{table}[htbp] 
  \caption{\bf Pricing Performance:  Out-of-Sample -  Expected Returns  ($T=120$)}
\medskip
\setlength{\tabcolsep}{3pt}
\begin{tabular}{lccccccccccc}
   \multicolumn{12}{c}{Out-of-Sample}   \\  \hline
   \# of factors:   &  $\tilde{r}$ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
  constant-PCA              &              -&  0.381  & 0.113 &  0.129 &  0.247 &  0.257 &  0.283 &  0.338 &  0.326 &  0.321 &  0.346 \\
  local-PCA (bal) &                     0.45    &  0.346 & 0.364 &  0.379 &  0.421 &  0.435 &  0.438 &  0.444 &  0.448 &  0.450 &  0.451 \\
local-PCA (unbal) &                                     0.851         & 0.841 &  0.842  &  0.844 &  0.848  &  0.848 &  0.849 &    0.849 &  0.850 &  0.850 &  0.851 \\  \hline
  \multicolumn{12}{c}{In-Sample}   \\  \hline
 \# of factors:   &  $\tilde{r} $ & $1$  &  $ 2 $ & $3$ & $4$ & $5$ & $ 6 $ & $7$   & $8$  & $9$ &  $10$ \\   \hline
 constant-PCA              &           - &   0.185 &0.0162 & -0.016 &-0.269 & -0.289& -0.307 & -0.353 & -0.351& -0.324 &  -0.325 \\
   local-PCA (bal) &     0.441 & 0.353 &  0.396 &  0.475 &  0.569 &  0.612 &  0.627 & 0.650 &  0.668 &  0.674 & 0.677 \\
local-PCA (unbal) &   0.885   &  0.842  &  0.841 &  0.852 &  0.865 &  0.869 &  0.872 &  0.876 &  0.878 &  0.880 &  0.885 \\ \hline
    \end{tabular}
        \parbox{6.3in}{\footnotesize{The table shows the cross-sectional correlations between the average returns and the model-implied expected return estimated by 
        constant-PCA (first row), local-PCA balanced (second row), and local-PCA unbalanced (third row), when the number of risk factors ranges from one to $10$  (columns $3-12$), with the second column referring to the
      (time-varying) estimated number of factors $\tilde{r}$, described in Section~\ref{emp_number_tv}. The average returns and model-implied expected returns are first evaluated over every rolling window of size $T=120$ and then averaged across all rolling windows. 
 Data are monthly from Jan 1980 until Dec 2019.}}
\label{Table3OAT120}
\end{table}



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